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A refinery produces two types of lubricating oil, type A and type B. To produce 1000 litres of type A oil requires two hours in process 1, two hours in process 2 and six hours in process 3. To produce 1000 litres of type B oil requires four hours in process 1, two hours in process 2 and three hours in process 3. The availability of the process 1 plant is twelve hours per day, that of process 2 is seven hours per day and that of process 3 twenty hours per day. The profit on 1000 litres of Type A is £40 and on Type B £50. The ompany wants to know how much of each type of oil to produce in order to maximise their profit per day, assuming that all the production can be sold. (a) What are the decision variables for this problem? (b) What is the objective function? (c) State the constraints as inequalities. (d) Sketch the feasible region, indicating the coordinates of each of the corner points. On your diagram, show the contour associated with a daily profit of £200. (e) Using this contour or otherwise (but NOT by using the Simplex Method), determine the optimum solution to the problem, justifying your approach. (f) How much of each type of oil should the Company produce each day? (g) The demand for Type A oil has risen greatly and the Company intends to increase its price. By how much can the profit on Type A oil be increased before the Company should produce this type of oil only? How much profit would the company make at this profit level. (h) By reorganising the workforce it is possible to increase the number of hours that the plant for process 2 is available. In order to maximise profit, how many extra hours should process 2 be made available? (carefully explain your reasoning) How much of each type of oil will the refinery then produce and what will be the company's profit?

Answer 1

(a) The Decision Variables are:

• Type A Production per day
• Type B Production per day

(b) The Objective function is to Maximize the Profit, where

• Profit = Type A Profit + Type B Profit
• Type A Profit = 40 * (Type A Production / 1000)   
• Type B Profit = 50 * (Type B Production / 1000)  

(c) subject to the following Constraints:

• Usage of P1 plant = ((2 * Type A Prodn / 1000) + (4 * Type B Prodn / 1000)) <= 12 hrs per day
• Usage of P2 plant = ((2 * Type A Prodn / 1000) + (2 * Type B Prodn / 1000)) <= 7 hrs per day
• Usage of P3 plant = ((6 * Type A Prodn / 1000) + (3 * Type B Prodn / 1000)) <= 20    hrs per day
• Type A Production >= 0
• Type B Production >= 0

(d) Feasible Region with below corner points and objective function:

Corner Points	OF
0, 0	0
0, 3000	150
3333, 0	133.32
1000, 2500	165
3166.67, 333.33	143.32